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#1 2010-12-19 07:31:27

Reuel
Member
Registered: 2010-11-28
Posts: 178

Curvilinear Coordinates

Hello. smile

I am interested in learning about curvilinear coordinates. I did a search on these forums and found only one use of the phrase and said incident was irrelevant to my question(s).

What I would like to know is, first of all, are curvilinear coordinates strictly a topic of tensor calculus? If so, need I first understand tensors in order to understand the topic of curvilinear coordinates?

My reason for asking is a curiosity in transforming a curve in space into a linear line on curve coordinate axes. I do not have any practical applications, I simply would like to learn about the concept.

How does ones "transform" a vector-valued space curve into a linear line on a curved coordinate system? Maple is one of my resources at hand.

I appreciate any and all answers. Thank you.

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#2 2010-12-19 08:18:00

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Curvilinear Coordinates

Hi Reuel;

Polar coordinates are an example of a curvilinear coordinate system. So are spherical and cylindrical. It is not a topic strictly for Tensor Calculus but you do go much deeper into it with tensors. There you can study all curvilinear systems even ones you make up.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2010-12-19 09:18:43

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Curvilinear Coordinates

bobbym,

Thank you for getting back to me so quickly! I appreciate and understand your answer. However, in polar and spherical coordinates, are shapes not still based on coordinate axes? At least, in my last class, shapes were still drawn in the octanes, simply defined by functions converted from rectangular to spherical coordinates.

My question was with regards to situations where space curves might be redefined as linear lines on curved coordinate systems. Unfortunately I don't really know well enough of what I am talking about to explain it better than that so I could try giving an example instead.

Suppose you have some basic vector-valued space curve in rectangular coordinates and with some parameter t, such as

and for whatever reason you want to describe a coordinate system where r(t) - in 3D space - may be described as a straight line on a curved "2D" surface - what I mean is, simply a curved coordinate system. r(t) would be a straight line on a curved surface.

Sorry ahead of time if I am asking illogical questions. smile

What sort of "transformation" would r(t) undergo?

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#4 2010-12-19 13:01:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Curvilinear Coordinates

Hi Reuel;

Sorry ahead of time if I am asking illogical questions.

My tensor calculus is too weak. If no one else can assist you then please try here, post it and get back and explain it to me. Sorry I can not help you more.

http://www.physicsforums.com/

There are a couple of vector gurus there who are stronger than I was even when I knew what I was doing.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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